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A slender rod, 0.240 m long, rotates with an angular speed of 8.80 rad/s about an axis through one end and perpendicular to the rod. The plane of rotation of the rod is perpendicular to a uniform magnetic field with a magnitude of 0.650 T. (a) What is the induced emf in the rod? (b) What is the potential difference between its ends? (c) Suppose instead the rod rotates at 8.80 rad/s about an axis through its center and perpendicular to the rod. In this case, what is the potential difference between the ends of the rod? Between the center of the rod and one end?

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A motional emf arises because the slender rod moves relative to $\vec{\pmb B}$. The complication is that different parts of the rod move at different speeds $v$, depending on their distance from the rotation axis. We'll address this by considering small segments of the rod and integrating their contributions to determine our target variable in part (a), the induced emf in the rod. Suppose the rod is rotating in the counterclockwise direction in a plane perpendicular to a uniform magnetic field $\vec{\pmb B}$ which points out of the page, as shown in Fig. 1a; Consider the small segment of the rod shown in red and labeled by its velocity vector $\vec{\pmb v}$. The magnetic force per unit charge on this segment is $\vec{\pmb v }\times \vec{\pmb B}$, which points radially outward to make a current flow radially outward from the fixed end of the rod to the free, rotating end. We can find the emf from each small rod element along its length by using $d\varepsilon = (\vec{\pmb v}\times\vec{\pmb B})\cdot d\vec{\pmb l}$ and then integrate to find the total emf. In part (b) we assume the rod poses no resistance to the flowing current due to the induced emf, so the potential difference is simply $|\varepsilon|$. In part (c) the rotation axis is through the rod's center (see Fig. 1b), so there is a resulting emf in each half of the rod and an induced current flows radially outward from the rod's center to both ends. By symmetry the induced emf in each half of the rod is the *same*, so we'll see that while the potential difference between the center of the rod and either end is the nonzero value $|\varepsilon|$ (calculated in the same way we described in part (a)), the potential difference between the rod's ends is zero.

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